Near-Feasible Solutions to Complex Stable Matching Problems
Gergely Csáji
TL;DR
The paper addresses NP-hard stable matching variants by proving that near-feasible stable solutions exist under small capacity perturbations. It introduces an iterative rounding framework that starts from a fractional stable solution obtained via Scarf's lemma and yields an integral, near-feasible stable solution across SHM, CA-CQ, and SMF, with a polynomial-time guarantee for the Stable Fixtures case. The key results bound capacity modifications: $|q'(v)-q(v)|\\le\\ell-1$ for SHM, $|q'(C_j)-q(C_j)|\\le\\ 2\\ell-1$ for CA-CQ, and $\max|c'(a)-c(a)|\\\le\\ k-1$ for SMF, while preserving stability. These findings have practical impact for market design, college admissions, and traffic and logistics networks, offering implementable stability with minimal constraint adjustments.
Abstract
In this paper, we demonstrate that in many NP-complete variants of the stable matching problem, such as the Stable Hypergraph Matching problem, the Stable Multicommodity Flow problem, and the College Admission problem with common quotas, a near-feasible stable solution - that is, a solution which is stable, but may slightly violate some capacities - always exists. Our results provide strong theoretical guarantees that even under complex constraints, stability can be restored with minimal capacity modifications. To achieve this, we present an iterative rounding algorithm that starts from a stable fractional solution and systematically adjusts capacities to ensure the existence of an integral stable solution. This approach leverages Scarf's algorithm to compute an initial fractional stable solution, which serves as the foundation for our rounding process. Notably, in the case of the Stable Fixtures problem, where a stable fractional matching can be computed efficiently, our method runs in polynomial time. These findings have significant practical implications for market design, college admissions, and other real-world allocation problems, where small adjustments to institutional constraints can guarantee stable and implementable outcomes.